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By Peter Barlow

Barlow P. An trouble-free research of the speculation of numbers (Cornell collage Library, 1811)(ISBN 1429700467)

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34) where the indices satisfy 1 ≤ i < j ≤ N . Stochastic processes ξ(t) in the above equations are taken to be all mutually independent Gaussian white noises characterized by the correlation functions: σ2 Eξ [ξi1 (t1 )ξi2 (t2 )] = 2Dδi1 ,i2 δ(t1 − t2 ), Eξ ξijσ1 (t1 )ξkl (t2 ) = Dδσ1 ,σ2 δi,k δj,l δ(t1 − t2 ). 35) As initial values xi (0), xij (0), yij (0) for each OU process we choose diagonal and (0) (0) (0) off-diagonal entries Hii , i = 1, . . , N and ReHi

KN (λ1 , λj ) . . . . . KN (λj , λ1 ) . . 21) ⎟ ⎟ ⎟ dλ1 . . dλj . ⎟ ⎠ In fact, the last expression can be written in a very compact form by noticing that it is just a Fredholm determinant det (I − KN ), where KN is a (finite −1 rank) integral operator with the kernel KN (λ, λ ) = N i=0 φi (λ)φi (λ ) acting on square-integrable functions on the interval λ ∈ (−L/2, L/2). 1) and consider, for k ≥ l ∞ e−N x2 2 ∞ hl (x)hk (x)dx = (−1)k −∞ −∞ x2 dk e−N 2 k dx x2 dk−1 dx hl (x) k−1 e−N 2 dx dx hl (x) = (−1)k+1 ∞ −∞ = ...

C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. , Proc. London Math. Soc. (3), 83 (2001), 532-562. -R. Chen, on the representation of a large even integer as a sum of a prime and a product of at most two primes, Kexue Tongbao, 17 (1966), 385-386. [3] H. Davenport, Multiplicative number theory, Graduate Texts in Mathematics, 74. (Springer-Verlag, New York-Berlin, 1980). B. Friedlander and H. Iwaniec, The polynomial X 2 + Y 4 captures its primes, Annals of Math. (2), 148 (1998), 945-1040.

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